VERIFYING A VECTOR IDENTITY USING LEVI-CIVITA
Bianca Rae B. Sambo
Hello physics enthusiast! I am BR, a graduate student in Physics in Mindanao State University – Iligan Institute of Technology. Hope my solution will be of use to you. Keep visiting this site!
The vector identity to be verified using Levi-Civita is,
where we define the following vectors as,
We start by expanding the left hand side of the identity.
Then we set i = j and apply the distributive property of the partial derivative.
Notice in the right hand side of the vector identity, we have the equations and . These two equations can be written as,
So we incorporate this to Equation 1 above.
Notice also that:
So plugging these equations to equation 2, we finally verified the vector identity because we get: